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Theorem 00lss 15699
Description: The empty structure has no subspaces (for use with fvco4i 5597). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss  |-  (/)  =  (
LSubSp `  (/) )

Proof of Theorem 00lss
StepHypRef Expression
1 noel 3459 . . 3  |-  -.  a  e.  (/)
2 base0 13185 . . . . . 6  |-  (/)  =  (
Base `  (/) )
3 eqid 2283 . . . . . 6  |-  ( LSubSp `  (/) )  =  ( LSubSp `
 (/) )
42, 3lssss 15694 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  C_  (/) )
5 ss0 3485 . . . . 5  |-  ( a 
C_  (/)  ->  a  =  (/) )
64, 5syl 15 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =  (/) )
73lssn0 15698 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =/=  (/) )
87neneqd 2462 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  ->  -.  a  =  (/) )
96, 8pm2.65i 165 . . 3  |-  -.  a  e.  ( LSubSp `  (/) )
101, 92false 339 . 2  |-  ( a  e.  (/)  <->  a  e.  (
LSubSp `  (/) ) )
1110eqriv 2280 1  |-  (/)  =  (
LSubSp `  (/) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   ` cfv 5255   LSubSpclss 15689
This theorem is referenced by:  00lsp  15738  lidlval  15946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-lss 15690
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